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I teach a course in real analysis and applications to partial differential equations in which I spend some weeks talking about Sobolev spaces. I have always used the symbol $C_0^\infty(\Omega)$ to denote the set of infinitely differentiable functions on the (open) subset $\Omega$ of ${\Bbb R}^n$ with compact support. A colleague once told me that this notation is bad and misleading, so that I should switch to $C_c^\infty(\Omega)$.

I admit that I could not understand his remark. After thiking about it, the only idea that came to my mind is that somebody might use the first notation to denote the set of those functions $u \colon \Omega \to \mathbb{R}^n$ that vanish at infinity (provided that $\Omega$ "contains" infinity). For example, when $\Omega$ is the whole space, we request that for every $\varepsilon>0$ there exists a compact subset $K$ such that $\sup_{x \in \complement K} |u(x)|<\varepsilon$.

My question is: is there any other reason why I should stop using $C_0^\infty$?

Siminore
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    I've always seen $C_c^\infty$ for test functions and the subscript zero used to indicate vanishing at infinity. – Keenan Kidwell Aug 26 '15 at 18:18
  • Strongly seconding @KeenanKidwell's comment, and your surmise that the subscript $0$ is for functions vanishing at infinity... – paul garrett Aug 26 '15 at 18:35
  • @KeenanKidwell The notation $C_0^\infty(\mathbb{R}^N)$ to denote the set of test functions $\mathcal{D}(\mathbb{R}^N)$ is not so rare. I agree that it is not consistent with the notation for sequences, where $c_{00}$ is the set of sequences with compact support. – Siminore Aug 27 '15 at 08:40
  • Dear @Siminore, I think $C_c^\infty$ could reasonably be called standard. – Keenan Kidwell Aug 27 '15 at 12:47
  • @KeenanKidwell Yes, with remarkable exceptions. For instance, Reed and Simon use $C_0^\infty$. – Siminore Aug 27 '15 at 16:55

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